\documentclass[a4paper,10pt]{article}
\usepackage{amsfonts}
\usepackage{amsmath}

%opening
\title{Dynamic Scheduling in Flow Shop with Switchable Jobs}
\author{Meng-Chang WANG$^{1,2}$, Yun-Qing RAO$^{1,2}$, Kun-Peng WANG$^{1,2}$\\
\begin{footnotesize}1.School of Mechanical Science \& Engineering, HUST,\end{footnotesize} \\ 
\begin{footnotesize}Wuhan, China, 430074\end{footnotesize}\\
\begin{footnotesize}2.State Key Lab of Digital Manufacturing Equipment \& Technology, \end{footnotesize}\\ 
\begin{footnotesize}Wuhan, China, 430074\end{footnotesize}}

\begin{document}

\maketitle

\begin{abstract}

\end{abstract}

\section{Introduction}

\section{Literature Review}

\section{Notations}
In this paper, we focus on 2-machine FSSJ problems especially, using following notations:
\begin{itemize}
\item $N_{R}=\sum_{i=1}^{M}N_{i}$: the total number of operation kinds in the shop, where $N_{i}$ is the number of operation kinds that machine $i$ can do, and $M=2$ in this paper;
 \item $C_{IN}$: the number of job (raw material) types to enter into the first machine;

\item $C_{OUT}$: the number of job (final product) types generated by the shop, in this paper, $C_{IN}<C_{OUT}$ especially;

\item $n_{0}$: the original number of jobs planned to be processed in the shop;

\item $S_{0}=\{J_{1},\dots,J_{n_{0}}\}$: the original job sequence, where $J_{j}$  is the $j$-th job in the sequence;

\item $O_{0}(j,i)$: the operation to be done on $J_{j}$ at machine $i$ according to $S_{0}$; 
%if for $i=1,\dots,k$, $O_{0}$
\item $O_{1}(c,i)$: the operation to be done at machine $i$, when the job's final product type is $c$; if $O_{1}(c_{1},i)=O_{1}(c_{2},i)$, $i=1,\dots,k-1$, $(k \leq M)$, but $O_{1}(c_{1},k)\not =O_{1}(c_{2},k)$, we say jobs with final product type $c_{1}$ and with type $c_{2}$ are \textbf{\textit{switchable before machine k}};

\item $R_{0}=\{(c_{1},d_{1}),\dots,(c_{n_{0}},d_{n_{0}})\}$: the original product order set, where $c_{j}(1 \leq c_{j} \leq C_{OUT})$ is the final product type of a job, and $d_{j}$ is the due date of the job;

\item $D_{0}(c,i)$: the $i$-th early due date of final product type $c$, according to $R_{0}$;

\item $n_{OUT}(c)$: the number of products of type $c$ in $R_{0}$;

\item $t_{C}(j)$: the completion time of $J_{j}$;

\item $D_{1}(c,i)$: the $i$-th early completion time of product type $c$;

\item $p_{r}$: the processing time of operation $r$, where $1 \leq r \leq N_{R}$;

\item $s(i,k,l)$: the set-up time of machine $i$ from operation $k$ to $l$, where $1 \le i \le M$, $1 \le k \le N_{R}$, $1 \le l \le N_{R}$;

\item $T_{s}=\sum_{i=1}^{M}\sum_{j=2}^{n_{0}}s(i,O_{0}(j-1,i),O_{0}(j,i))$: the total set-up time according to $S_{0}$ 
%, and \[T_{s}=\sum_{i=1}^{M}\sum_{j=2}^{n_{0}}s(i,O_{0}(j-1,i),O_{0}(j,i))\]
;

\item $\Delta T_{d}=\sum_{c=1}^{C_{OUT}}\sum_{i=1}^{n_{OUT}(c)} \vert D_{1}(c,i)-D_{0}(c,i) \vert$: the deviation of completion time and due time;

\item $T_{A}$: the time when the accident happens, which could be emergent orders, work-piece quality rejection, or due date changes, and $0<T_{A}<t_{c}(n_{0})$ in this paper;

\item $R^{'}=(c^{'},d^{'})$: an emergent job, where $c^{'}$ is the final product type and $d^{'}$ is its due date;

\item $Q_{j}$: an rejected work-piece for quality reasons, which is also the $j$-th job in $S_{0}$;

\item $D^{'}(c,i)$: the new due date of the $i$-th early due date of final product type $c$ in $R_{0}$;
\end{itemize}

\begin{LARGE}\textbf{\textit{Triplet expression}}:\end{LARGE}

For example, $F2\vert T_{A},R^{'} \vert \Delta T_{d}$ means a 2-machine flow shop with a rush job $R^{'}$ comes at time $T_{A}$.

\begin{center}
% use packages: array
\begin{tabular}{lll}
No & Model &  Comment\\ 
1 & $F2\vert T_{A},R^{'} \vert \Delta T_{d}$ & section 4.1 \\ 
2 & $F2\vert T_{A},R^{'} \vert \Delta T_{s}$ & section 4.2 \\
\end{tabular}
\end{center}


\section{Algorithms}
In this section, we provide dynamic scheduling algorithms for different cases, with the assumption that an \textit{optimal} squence have been generated as the original job sequence $S_{0}$.
















\end{document}
